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1st May 2006, 01:10 AM  #11 
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In my world, the resonances are the eigenvalues of the system model. A BR is a 2 degree of freedom system, assuming lumped parameter modeling, hence two resonances. A linear combination of the eigenvectors, or mode shapes, can be used to express any motion of the BR system.

1st May 2006, 03:45 PM  #12  
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1st May 2006, 04:12 PM  #13 
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The BR/driver system does not have a Helmholtz resonance.
The BR/driver system has two new resonances one above and one below the box Helmholtz resonance. The mode shape of the lower resonance has the driver mass moving into the enclosure and the port air mass moving out of the enclosure ( in phase hence 24 dB/octave rolloff). The second mode shape has the driver mass moving out of the enclosure and the port air mass moving out of the enclosure (out of phase stretching the box "spring"). In between these resonances, the mode shapes combine which produces minimal motion at the driver and maximum motion at the port, people mistake this condition for the resonance of the box (Helmholtz resonance) but it is not a true resonant condition for the BR system. It is only confusing if you do not have a firm grasp on the math and are considereing only cause and effect types of observations. Once the math is set in your thinking, it makes perfect sense and a clearer insight is obtained into what is going on in a BR system (or TL system) and what options are available for influencing the overall system response. 
1st May 2006, 04:55 PM  #14 
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Thanks a lot MJK, i think i got the point.It's much more complicated than i thought.

1st May 2006, 07:12 PM  #15 
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Ok, just to set my mind straight on this (I am not used to talking about eigenvalues even though I think I know roughly what they are.):
If you calculate the transfer function of any system, there will be poles and zeroes. The poles are inherent to the system, but the zeroes depend on where you excite it and where you measure. These poles are closely linked to the eigenvalues, as I understand it. The BR system has two pole pairs, since it is fourth order, and these two pole pairs correspond to the eigenvalues? Am I right? But Hmm... That would make the two eigenvalues of a butterworth design occur at the same frequency (but different Q) ie the helmholtz frequency. Is there such a thing as a Q value for an eigenvalue? Hmm. Interesting this, there are sooo many ways of looking at loudspeakers 
1st May 2006, 08:12 PM  #16 
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Oh no ....... poles and zeros. I am a Mechanical Engineer so my understanding of poles and zeros is minimal. But I think you are correct that the BR system will have four poles (from the denominator of the transfer function). I also believe that the four poles are really two complex conjugates of the form
p1 = a + jb p2 = a  jb p3 = c + jd p4 = c  jd where the a and c are sometimes labeled as time constants and the b and d are sometimes labeled as damped natural frequencies. The exponential decay damping is derived from a and c while the resonant frequencies are calculated from b and d. Unfortunately that is about the limit of my understanding (at least what I remember from 25+ years ago) of poles, zeros, and transfer functions. Most of the math/analysis work I do now is numerical methods and finite element based so the plotting of a system's response from poles and zeros plots is way back in my distant undergraduate days past. 
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